

NAVSO P3679: Producibility Measurement Guidelines/Methodologies 
 

4.3 The Basis of Producibility Metrics
To fully address the issue of producibility measurement and
numerically optimizing product and process designs in accordance with known
production capabilities, key points associated with modern probability theory
must first be discussed.
The Producibility Game
Since the issue of producibility involves the ideas of
manufacturing confidence and risk, it is reasonable to assert that the concepts
underlying probability should serve as the foundation of any measurement scheme.
For example, imagine a pair of "manufacturing dice" and a customer requirement
which only allows those combinations which yield a 3, 4, 5,...,or 11. In this
instance, a 2 or 12 represents a nonconformance to standard  or a quality
defect. The question to what extent the customer will be satisfied is therefore
statistically stated. "What is the probability of not rolling a 2 or 12?" To
answer the latter question, some fundamental probability theory must be
applied.
The likelihood of some event A may be given
by P(A). If some event A is independent of some other event, say B, the
probability of both A and B occurring is P(A) x P(B). In other words, the
joint probability of A and B is multiplicative. Since a single die has six
sides, the random chance probability that any given side will be face up is
1/6 = .1667 because only one side can be up at any given time, and there are a
total of six possibilities. The random chance probability is thus that any
particular side will be face up is 16.67%. Given that there is a pair of dice,
the probability of rolling two 1s would be .1667 x .1667 = .0278, or 2.78%.
The same reasoning would also hold true for the possibility of two 6s. This
may be seen by studying the exhaustive combinations given by a pair of dice
(Figure 42).
Because the occurrence of a 2 or 12 can not
happen concurrently, the two outcomes are mutually exclusive of each other. In
other words, a 2 and 12 cannot occur at the same time  they are restricted
from occurring together. The events of concern are mutually exclusive. Any
time that the two events A and B are mutually exclusive, the probability of
event A or B occurring may be given by summing their individual probabilities,
P(A) + P(B). Therefore, the total probability of not meeting the customer's
standards, with respect to the manufacturing example, would be .0278 + .0278 =
.0556, or 5.56%. This represents the risk of nonconformance. It can be
reasoned that the likelihood of yielding a 3, 4, 5...., or 11 may be given by
1  .0556 = .9444. In the context of the example, this would reveal that the
likelihood of customer satisfaction is 94.44%. (Additional insight into these
ideas may be seen by examining Figure 43, using Figure 42 as a frame of reference.)
Now suppose that there are two sets of dice
and that the first set of dice is the gaming vehicle for the previously
mentioned specification (the customer has defined a 2 or 12 as a defect). As
already indicated, the yield related to the first specification was determined
to be 94.44%. Now assume that the second set of dice is related to a different
specification. In this instance, the second specification is given by the
combinatorial possibilities yielding a 4, 5, 6...., or 10  or a 2, 3, 11, or
12 would be considered defective. By the same statistical reasoning process,
it would be determined that the expected yield, as it applies to the second
specification, would be 83.33%.
The question of what the probability of simultaneously
meeting both specifications is could then be asked. When two events are
independent, the joint probability is multiplicative. In this instance, the
likelihood of conformance to both specifications can be described by [ 1P(A)]
x [ 1P(B)]. As applied to the example situation, the probability of picking
up both sets of dice and rolling a "good" number would be .9444 x .8333 =
.7870. This would be to say that there is a 78.70% chance of simultaneously
producing numbers which conform to both specifications. In short, the
likelihood of success, or in this case first time yield, is a direct measure
of producibilityif the joint or "rolled" yield is high, producibility is
high; otherwise, producibility is low.
Probability and Product Parameters
The ideas behind the gaming example to an
actual product parameter can now be generalized. To do so, suppose that there
is some bilateral product performance specification related to parameter
Y_{1}. Here again, Y_{1} can be whatever  such as the length
of a mechanical part, voltage in a circuit, or resistance of a material. Also
assume that the underlying distribution is normal. If event A represents a
nonconformance to standard and the probability of encountering an A is .27%,
then the "first time" yield of parameter Y_{1}
would
be 1.0000  .0027 = .9973, or 99.73%.
If the yield of some other independent characteristic
Y_{2} was .8845, the joint probability of conformance to specification
with respect to Y_{1} and Y_{2} would be .9973 X .8845 =
.8821, or simply 88.2%. The joint yield is also called "rolledthroughput
yield," and designated as Y_{RT<
/SUB >
because the first time yield
values are mathematically rolled into a single value, where that single value is
indicative of the likelihood of nonincidental
throughput.}




 
 